Find the probability mass function of Y=sum_{i=1}^{N} X_i where X_i∼Geometric(a) and N∼Geometric(b). I attempted to do this by finding the probability generating function of Y and comparing it to known probability generating functions to take advantage of the uniqueness property. (In my searches online, it sounds like I should find that Y∼Geometric(ab).)

Sonia Rowland 2022-10-01 Answered
Geometric sum of geometric random variables
I am looking to find the probability mass function of Y = i = 1 N X i where X i Geometric ( a ) and N Geometric ( b ). I attempted to do this by finding the probability generating function of Y and comparing it to known probability generating functions to take advantage of the uniqueness property. (In my searches online, it sounds like I should find that Y Geometric ( a b ).)
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Answers (1)

Cody Petty
Answered 2022-10-02 Author has 6 answers
Step 1
Assume that (1) P [ X i = k ] = p ( 1 p ) k 1 , k = 1 , 2 ,
and P [ N = n ] = r ( 1 r ) n 1 for r = 1 , 2 , . Then: P [ Y = m ] = n 1 P [ N = n ] P [ X 1 + + X n = m ] (2) = r 1 r n 1 ( 1 r ) n ( p 1 p ) n ( 1 p ) m r ( m , n )
Where: r ( m , n ) = # { ( a 1 , , a n ) : a i N > 0 , a 1 + + a n = m } = [ x m ] ( x + x 2 + x 3 + ) n (3) = [ x m ] x n ( 1 x ) n = ( m 1 n 1 )
Step 2
Hence: P [ Y = m ] = r p ( 1 p ) m 1 n 1 ( m 1 n 1 ) ( p p r 1 p ) n 1 = r p ( 1 p ) m 1 ( 1 + p p r 1 p ) m 1 (4) = p r ( 1 p r ) m 1
proving your claim for geometric distributions supported on 1,2,…. There is little to change in order to deal with geometric distributions supported on 0,1,…, too.
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